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Mathematics 10 Learner’s Material Unit 4

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Mathematics 10 Learner’s Material Unit 4

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Mathematics 10 Learner’s Material Unit 4

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Mathematics 10 Learner’s Material Unit 4

1. 1. D EPED C O PY 10 Mathematics Department of Education Republic of the Philippines This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at action@deped.gov.ph. We value your feedback and recommendations. Learner’s Module Unit 4 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
3. 3. D EPED C O PY Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
4. 4. D EPED C O PY Unit 4 Module 8: Measures of Position............................................................ 355 Lessons and Coverage........................................................................ 357 Module Map......................................................................................... 357 Pre-Assessment .................................................................................. 358 Learning Goals and Targets ................................................................ 361 Lesson 1: Measures of Position for Ungrouped Data................................ 362 Activity 1.................................................................................... 362 Activity 2.................................................................................... 363 Activity 3.................................................................................... 363 Activity 4.................................................................................... 364 Activity 5.................................................................................... 369 Activity 6.................................................................................... 371 Activity 7.................................................................................... 371 Activity 8.................................................................................... 372 Activity 9.................................................................................... 372 Activity 10.................................................................................. 375 Activity 11.................................................................................. 375 Activity 12.................................................................................. 377 Activity 13.................................................................................. 378 Activity 14.................................................................................. 378 Activity 15.................................................................................. 379 Activity 16.................................................................................. 379 Activity 17.................................................................................. 380 Summary/Synthesis/Generalization........................................................... 382 Lesson 2: Measures of Position for Grouped Data.................................... 383 Activity 1.................................................................................... 383 Activity 2.................................................................................... 384 Activity 3.................................................................................... 394 Activity 4.................................................................................... 395 Activity 5.................................................................................... 396 Activity 6.................................................................................... 396 Activity 7.................................................................................... 397 Activity 8.................................................................................... 398 Activity 9.................................................................................... 398 Activity 10.................................................................................. 401 Activity 11.................................................................................. 401 Summary/Synthesis/Generalization........................................................... 402 Glossary of Terms ...................................................................................... 403 References and Website Links Used in this Module ................................. 403 Table of Contents All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
7. 7. D EPED C O PY 357 II. LESSONS AND COVERAGE Lesson 1 – Measures of Position for Ungrouped Data Lesson 2 – Measures of Position for Grouped Data In this lesson, you will learn to: Lesson 1 Lesson 2  illustrate the following measures of position: quartiles, deciles, and percentiles.  calculate specified measure of position (e.g., 90th percentile) of a set of data.  interpret measures of position.  solve problems involving measures of position.  formulate statistical mini-research.  use appropriate measures of position and other statistical methods in analyzing and interpreting research data. Here is a simple map of the lessons in this entire module. Study Tips To do well in this particular topic, you need to remember and do the following: 1. Study each part of the module carefully. 2. Take note of all the formulas given in each lesson. 3. Have your own scientific calculator. Make sure you are familiar with the keys and functions of your calculator. Solving Real-Life Problems Measures of Position Ungrouped Data Decile PercentileQuartile Grouped Data Decile PercentileQuartile All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
8. 8. D EPED C O PY 358 III. PRE-ASSESSMENT Part I. Find out how much you already know about this module. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module. 1. The median score is also the _____________. A. 75th percentile C. 3rd decile B. 5th decile D. 1st quartile 2. When a distribution is divided into hundred equal parts, each score point that describes the distribution is called a ___________. A. percentile C. quartile B. decile D. median 3. The lower quartile is equal to ______________. A. 50th percentile C. 2nd decile B. 25th percentile D. 3rd quartile 4. Rochelle got a score of 55 which is equivalent to 70th percentile in a mathematics test. Which of the following is NOT true? A. She scored above 70% of her classmates. B. Thirty percent of the class got scores of 55 and above. C. If the passing mark is the first quartile, she passed the test. D. Her score is below the 5th decile. 5. In the set of scores: 14, 17, 10, 22, 19, 24, 8, 12, and 19, the median score is _______. A. 17 C. 15 B. 16 D. 13 6. In a 70-item test, Melody got a score of 50 which is the third quartile. This means that: A. she got the highest score. B. her score is higher than 25% of her classmates. C. she surpassed 75% of her classmates. D. seventy-five percent of the class did not pass the test. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
9. 9. D EPED C O PY 359 7. The 1st quartile of the ages of 250 fourth year students is 16 years old. Which of the following statements is true? A. Most of the students are below 16 years old. B. Seventy-five percent of the students are 16 years old and above. C. Twenty-five percent of the students are 16 years old. D. One hundred fifty students are younger than 16 years. 8. In a 100-item test, the passing mark is the 3rd quartile. What does it imply? A. The students should answer at least 75 items correctly to pass the test. B. The students should answer at least 50 items correctly to pass the test. C. The students should answer at most 75 items correctly to pass the test. D. The students should answer at most 50 items correctly to pass the test. 9. In a group of 55 examinees taking the 50-item test, Rachel obtained a score of 38. This implies that her score is ______________. A. below the 50th percentile C. the 55th percentile B. at the upper quartile D. below the 3rd decile 10. Consider the score distribution of 15 students given below: 83 72 87 79 82 77 80 73 86 81 79 82 79 74 74 The mean in the given score distribution of 15 students can also be interpreted as ______. A. seven students scored higher than 79. B. seven students scored lower than 79. C. seven students scored lower than 79 and seven students scored higher than 79. D. fourteen students scored lower than 79. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
10. 10. D EPED C O PY 360 For items 11 to 14, refer to table A below. Table A Score Frequency Cumulative Frequency Cumulative Percentage (%) 40-45 6 18 100.00 35-39 5 12 66.67 30-34 3 7 38.89 25-29 4 4 22.22 11.In solving for the 60th percentile, the lower boundary is ___. A. 34 C. 39 B. 34.5 D. 39.5 12.What cumulative frequency should be used in solving for the 35th percentile? A. 4 C. 12 B. 7 D. 18 13.The 45th percentile is ________. A. 33.4 C. 30.8 B. 32.7 D. 35.6 14.The 50th percentile is _____. A. 36.0 C. 36.5 B. 37.0 D. 37.5 Part II. Read and understand the situation below, then answer or perform what is asked. (6 points) Jefferson, your classmate, who is also an SK Chairman in Barangay Cut-Cot, organized a Run for a Cause activity, titled FUN RUN. He informed your school principal to motivate students to join the said FUN RUN. Conduct a mini-research or a simple research study on the students’ performance on the number of minutes it took them to reach the finish line. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
12. 12. D EPED C O PY 362 Let us start our study of this module by first reviewing the concept of median, which is one of the concepts needed in the study of this module. Discuss the answers to the questions below with a partner. The midpoint between two numbers x and y on the real number line is 2 x y . A B C    x 2 x y y 1. Find the coordinates of the midpoint (Q1) of AB in terms of x and y. A Q1 B    x 2 x y 2. Find the coordinates of the midpoint (Q2) of BC in terms of x and y. B Q2 C    2 x y y 3. In the given example, AC represents a distribution. What does point B represent in the distribution? Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
14. 14. D EPED C O PY 364 This part of the module enables you to understand quantiles in a set of ungrouped data. The activities in this section will help you answer the question, What are the ways to determine the measure of position in a given set of data? The understanding that you will gain in doing these activities will help you understand measures of position. 1. You are the fourth tallest student in a group of 10. If you are the 4th tallest student, therefore 6 students are shorter than you. It also means that 60% of the students are shorter than you. If you are the 8th tallest student in a group of 10, how many percent of the students are shorter than you? _________________________________ 2. A group of students obtained the following scores in their statistics quiz: 8 , 2 , 5 , 4 , 8 , 5 , 7 , 1 , 3 , 6 , 9 Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
15. 15. D EPED C O PY 365 Middle Quartile is also the_______ First, arrange the scores in ascending order: 1 , 2 , 3 , 4 , 5 , 5 , 6 , 7 , 8 , 8 , 9 Observe how the lower quartile (Q1), middle quartile (Q2), and upper quartile (Q3) of the scores are obtained. Complete the statements below: The first quartile 3 is obtained by ____________________________. (observe the position of 3 from 1 to 5) The second quartile 5 is obtained by _________________________ . (observe the position of 5 from 1 to 9) The third quartile 8 is obtained by ___________________________ . (observe the position of 8 from 6 to 9). 3. The scores of 10 students in a Mathematics seatwork are: 7 , 4 , 8 , 9 , 3 , 6 , 7 , 4 , 5 , 8 Arrange the scores in ascending order: 3 , 4 , 4 , 5 , 6 , 7 , 7 , 8 , 8 , 9 Discuss with your group mates: a. your observations about the quartile. b. how each value was obtained. c. your generalizations regarding your observations. Q3 Upper quartile Q1 Lower quartile Q2 Middle quartile Q3 Upper quartile Q1 Lower quartile Q2 6 7 6.5 2   All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
16. 16. D EPED C O PY 366 Let us take a closer look at the quartiles. The Quartile for Ungrouped Data The quartiles are the score points which divide a distribution into four equal parts. Twenty-five percent (25%) of the distribution are below the first quartile, fifty percent (50%) are below the second quartile, and seventy-five percent (75%) are below the third quartile. 1 Q is called the lower quartile and 3 Q is the upper quartile. 1 Q < 2 Q < 3 Q , where 2 Q is nothing but the median. The difference between 3 Q and 1 Q is the interquartile range. Since the second quartile is equal to the median, the steps in the computation of median by identifying the median class is the same as the steps in identifying the Q1 class and the Q3 class. a. 25% of the data has a value ≤ Q1 b. 50% of the data has a value ≤ X or Q2 c. 75% of the data has a value ≤ Q3 Example 1. The owner of a coffee shop recorded the number of customers who came into his café each hour in a day. The results were 14, 10, 12, 9, 17, 5, 8, 9, 14, 10, and 11. Find the lower quartile and upper quartile of the data. Solution:  In ascending order, the data are 5, 8, 9, 9, 10, 10, 11, 12, 14, 14, 17  The least value in the data is 5 and the greatest value in the data is 17.  The middle value in the data is 10.  The lower quartile is the value that is between the middle value and the least value in the data set.  So, the lower quartile is 9.  The upper quartile is the value that is between the middle value and the greatest value in the data set.  So, the upper quartile is 14. Q1 Q2 Q3 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
17. 17. D EPED C O PY 367 Example 2. Find the average of the lower quartile and the upper quartile of the data. Component Quantity hard disk 290 monitors 370 keyboards 260 mouse 180 speakers 430 Solution:  In increasing order, the data are 180, 260, 290, 370, 430.  The least value of the data is 180 and the greatest value of the data is 430.  The middle value of the data is 290.  The lower quartile is the value that is between the least value and the middle value.  So, the lower quartile is 260.  The upper quartile is the value that is between the greatest value and the middle value.  So, the upper quartile is 370.  The average of the lower quartile and the higher quartile is equal to 315. Example 3. The lower quartile of a data set is the 8th data value. How many data values are there in the data set? Solution:  The lower quartile is the median data value of the lower half of the data set.  So, there are 7 data values before and after the lower quartile.  So, the number of data values in the lower half is equal to 7+7+1.  The number of values in the data set is equal to lower half + upper half + 1.  The number of values in the lower and upper halves are equal.  Formula: 15+15+1=31  So, the data set contains 31 data values. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
18. 18. D EPED C O PY 368 Another solution: 1 4 (n + 1) = 8 n + 1 = 32 n = 31 Example 4. Mendenhall and Sincich Method. Using Statistics for Engineering and the Sciences, define a different method of finding quartile values. To apply their method on a data set with n elements, first calculate: Lower Quartile (L) = Position of  1 1 1 4 Q n  and round to the nearest integer. If L falls halfway between two integers, round up. The Lth element is the lower quartile value (Q1). Next calculate: Upper Quartile (U) = Position of   3 3 1 4 Q n and round to the nearest integer. If U falls halfway between two integers, round down. The Uth element is the upper quartile value (Q3). So for our example data set: {1, 3 , 7, 7, 16 , 21, 27, 30 , 31} and n = 9. To find Q1, locate its position using the formula  1 1 4 n  and round off to the nearest integer. Position of   1 1 1 4 Q n 1 4  (9 + 1)  1 4 (10)  2.5 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
19. 19. D EPED C O PY 369 The computed value 2.5 becomes 3 after rounding up. The lower quartile value (Q1) is the 3rd data element, so Q1 = 7. Similarly: Position of  3 3 1 4 Q n   3 9 1 4    3 10 4  = 7.5 The computed value 7.5 becomes 7 after rounding down. The upper quartile value (Q3) is the 7th data element, so Q3 = 27. Using this method, the upper quartile (Q3) and lower quartile (Q1) values are always two of the data elements. Find the first quartile (Q1), second quartile (Q2), and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Mendenhall and Sincich Method. 4 9 7 14 10 8 12 15 6 11 Example 5. Find the first quartile (Q1) and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Linear Interpolation. 1 27 16 7 31 7 30 3 21 Solution: a. First, arrange the scores in ascending order. 1 3 7 7 16 21 27 30 31 Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
20. 20. D EPED C O PY 370 b. Second, locate the position of the score in the distribution. Position of  1 1 1 4 Q n   1 9 1 4    1 10 4  = 2.5 Since the result is a decimal number, interpolation is needed. c. Third, interpolate the value to obtain the 1st quartile. Steps of Interpolation Step 1: Subtract the 2nd data from the 3rd data. 7 – 3 = 4 Step 2: Multiply the result by the decimal part obtained in the second step (Position of Q1). 4(0.5) = 2 Step 3: Add the result in step 2, to the 2nd or smaller number. 3 + 2 = 5 Therefore, the value of Q1 = 5. Solution: a. First, arrange the scores in ascending order. 1 3 7 7 16 21 27 30 31 b. Second, locate the position of the score in the distribution. Position of  3 3 1 4 Q n   3 9 1 4   3 10 4  = 7.5 Since the result is a decimal number, interpolation is needed. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
21. 21. D EPED C O PY 371 c. Third, interpolate the value to obtain the 3rd quartile. Steps of Interpolation Step 1: Subtract the 7th data from the 8th data. 30 - 27 = 3 Step 2: Multiply the result by the decimal part obtained in the third step (Position of Q3). 3(0.5) = 1.5 Step 3: Add the result in step 2, (1.5), to the 7th or smaller number. 27 + 1.5 = 28.5 Therefore, the value of Q3 = 28.5 Note: As we can see, these methods sometimes (but not always) produce the same results. Find the first quartile (Q1), second quartile (Q2), and the third quartile (Q3), given the scores of 10 students in their Mathematics activity using Linear Interpolation. 4 9 7 14 10 8 12 15 6 11 Albert has an assignment to ask at random 10 students in their school about their ages. The data are given in the table below. Name Age Name Age Ana 10 Tony 11 Ira 13 Lito 14 Susan 14 Christian 13 Antonette 13 Michael 15 Gladys 15 Dennis 12 Activity 7: Activity 6: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
22. 22. D EPED C O PY 372 1. What is Q1, Q2, and Q3 of their ages? 2. How many students belong to Q1, Q2, and Q3 in terms of their ages? 3. Have you realized the process of finding quartiles while doing the activity? Aqua Running has been promoted as a method for cardiovascular conditioning for the injured athlete as well as for others who desire a low impact aerobic workout. A study reported in the Journal of Sports Medicine investigated the relationship between exercise cadence and heart rate by measuring the heart rates of 20 healthy volunteers at a cadence of 48 cycles per minute (a cycle consisted of two steps). The data are listed here: 87 109 79 80 96 95 90 92 96 98 101 91 78 112 94 98 94 107 81 96 Find the lower and upper quartiles of the data. Consider the following nicotine levels of 40 smokers: 0 87 173 253 1 103 173 265 1 112 198 266 3 121 208 277 17 123 210 284 32 130 222 289 35 131 227 290 44 149 234 313 48 164 245 477 86 167 250 491 Find the lower and upper quartiles of the data. Activity 9: Activity 8: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
23. 23. D EPED C O PY 373 The Deciles for Ungrouped Data The deciles are the nine score points which divide a distribution into ten equal parts. They are deciles and are denoted as D1, D2, D3,…, D9. They are computed in the same way that the quartiles are calculated. Example 6. Find the 3rd decile or D3 of the following test scores of a random sample of ten students: 35 , 42 , 40 , 28 , 15 , 23 , 33 , 20 , 18 and 28. Solution: First, arrange the scores in ascending order. 15 18 20 23 28 28 33 35 40 42 Steps to find decile value on a data with n elements: To find its D3 position, use the formula  3 1 10 n  and round off to the nearest integer. Position of  3 3 10 1 10 D    3 11 10  33 10  = 3.3 ≈ 3 D3 is the 3rd element. Therefore, D3 = 20. D1 D2 D3 D4 D5 D6 D7 D8 D9 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
24. 24. D EPED C O PY 374 Example 7 1. Mrs. Labonete gave a test to her students in Statistics. The students finished their test in 35 minutes. This time is the 2.5th decile of the allotted time. What does this mean? Explanation: This means that 25% of the learners finished the test. A low quartile considered good, because it means the students finished the test in a short period of time. 2. Anthony is a secretary in one big company in Metro Manila. His salary is in the 7th decile. Should Anthony be glad about his salary or not? Explain your answer. Solution: 70% of the employees receive a salary that is less than or equal to his salary and 30% of the employees receive a salary that is greater than his salary. Anthony should be pleased with his salary. D2.5 35 minutes All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
25. 25. D EPED C O PY 375 Mrs. Marasigan is a veterinarian. One morning, she asked her secretary to record the service time for 15 customers. The following are service times in minutes. 20, 35, 55, 28, 46, 32, 25, 56, 55, 28, 37, 60, 47, 52, 17 Find the value of the 2nd decile, 6th decile, and 8th decile. After studying several discussions, examples, and activities, it will be good for you to look back and check if there are still aspects which you find confusing and hard. You are now ready to answer questions like: How can the position of a certain value in a given set of data be described and used in solving real-life problems? Given 50 multiple-choice items in their final test in Mathematics, the scores of 30 students are the following: 23 38 28 46 22 20 18 34 36 35 45 48 16 22 27 25 29 31 30 25 44 21 18 43 21 26 37 29 13 37 Calculate the following using the given data. 1. Q1 4. D2 2. Q2 5. D3 3. Q3 Activity 11: Activity 10: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
26. 26. D EPED C O PY 376 The Percentile for Ungrouped Data The percentiles are the ninety-nine score points which divide a distribution into one hundred equal parts, so that each part represents the data set. It is used to characterize values according to the percentage below them. For example, the first percentile (P1) separates the lowest 1% from the other 99%, the second percentile (P2) separates the lowest 2% from the other 98%, and so on. The percentiles determine the value for 1%, 2%,…, and 99% of the data. P30 or 30th percentile of the data means 30% of the data have values less than or equal to P30. The 1st decile is the 10th percentile (P10). It means 10% of the data is less than or equal to the value of P10 or D1, and so on. Example 8 Find the 30th percentile or P30 of the following test scores of a random sample of ten students: 35, 42, 40, 28, 15, 23, 33, 20, 18, and 28. Solution: Arrange the scores from the lowest to the highest. 15 18 20 23 28 28 33 35 40 42 Q1 Q2 Q3 P25 P50 P75 P10 P20 P30 P40 P50 P60 P70 P80 P90 D1 D2 D3 D4 D5 D6 D7 D8 D9 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
27. 27. D EPED C O PY 377 Steps to find percentile value on a data with n elements: To find its P30 position use the formula  1 100 k n and round off to the nearest integer. Position of   30 30 10 1 100 P    30 11 100  300 100  = 3.3 = 3.3 ≈ 3 P30 is the 3rd element. Therefore, P30 = 20. The scores of Miss World candidates from seven judges were recorded as follows: 8.45, 9.20, 8.56, 9.13, 8.67, 8.85, and 9.17. 1. Find the 60th percentile or P60 of the judges’ scores. 2. What is the P35 of the judges’ scores? Activity 12: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
28. 28. D EPED C O PY 378 Given a test in Calculus, the 75th percentile score is 15. What does it mean? What is its measure of position in relation to the other data? Interpret the result and justify. Complete the Cross Quantile Puzzle by finding the specified measures of position. Use linear interpolation. (In filling the boxes, disregard the decimal point. For example, 14.3 should be written as . Given: Scores 5, 7, 12, 14, 15, 22, 25, 30, 36, 42, 53, 65 1 2 3 4 5 6 7 8 9 341 Activity 14: Activity 13: Across 2. D7 4.  65 1 100 n 8.  90 1 100 n  9. P9 Down 1. Q2 3.  90 1 100 n  5. P40 6. P52 7. P54 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
29. 29. D EPED C O PY 379 This section of the module will test your understanding of the different measures of position by applying it to real-life situations. To demonstrate and apply your knowledge, you will be given a practical task specifically in the field of business and social sciences. Write each step in finding the position / location in the given set of data using the cloud below. Add or delete clouds, if necessary. A total of 8000 people visited a shopping mall over 12 hours. Estimate the third quartile (when 75% of the visitors had arrived). Estimate the 40th percentile (when 40% of the visitors had arrived). Time (hours) People 2 450 4 1500 6 2300 8 5700 10 6850 12 8000 Activity 16: Activity 15: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
30. 30. D EPED C O PY 380 Create a scenario of the task in paragraph form incorporating GRASPS. Goal: Make your own criteria in choosing the Cleanest Classroom Role: Students by Section Audience: The School Administration and Supreme Student Government Officers Situation: The SSG Officers will reward a certificate of recognition to those who will rank 1st based on the given standards. Product /Performance: Criteria Standards: Understanding of task, completion of task, communication of findings, group process Activity 17: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
32. 32. D EPED C O PY 382 SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about measures of position for ungrouped data. The lesson provided you with opportunities to illustrate and compute for quartiles, deciles, and percentiles of ungrouped data. You were also given the opportunity to formulate and solve real-life problems involving measures of position. You have learned the following: Quartile for Ungrouped Data Position of  1 4k k Q n  Decile for Ungrouped Data Position of  1 10k k D n  Percentile for Ungrouped Data Position of  1 100k k P n  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
33. 33. D EPED C O PY 383 To check your readiness for the next topic, review the previous lessons. These will help you in the study of measure of position for grouped data. As you study the module, you may answer the question: How are measures of position for grouped data used in real-life situations? Do and accomplish the activities with your partner. The following are scores of ten students in their 40-item quiz. 34 23 15 27 36 21 20 13 33 25 1. What are the scores of the students which are less than or equal to 25% of the data? ______________________________________________________ 2. What are the scores of the students which are less than or equal to 65% of the data? ______________________________________________________ 3. What are the scores of the students which are less than or equal to 8% of the data? ______________________________________________________ Activity 1: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
34. 34. D EPED C O PY 384 Use your scientific calculator to answer the following questions. Do this activity as fast as you can. 1. The bank manager observes the bank deposits in one specific day are as follows: 1150 5000 6500 1000 8500 9000 1200 1750 1100 4500 750 1500 1600 11 000 12 500 7000 9500 1200 13 500 1400 Find the 75th percentile. 2. The weights of the students in a class are the following: 69, 70, 75, 66, 83, 88, 66, 63, 61, 68, 73, 57, 52, 58, and 77. Compute the 15th percentile. 3. Mr. Mel Santiago is the sales manager of JERRY’S Bookstore. He has 40 sales staff members who visit college professors all over the Philippines. Each Saturday morning, he requires his sales staff to send him a report. This report includes, among other things, the number of professors visited during the previous week. Listed below, ordered from smallest to largest, are the number of professors visited last week. 38 40 41 45 48 48 50 50 51 51 52 52 53 54 55 55 55 56 56 57 59 59 59 62 62 62 63 64 65 66 66 67 67 69 69 71 77 78 79 79 Determine the following. a. 3rd quartile b. 9th decile c. 33rd percentile Activity 2: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
35. 35. D EPED C O PY 385 1Q f Did you find the previous activities easy? Were you able to answer it? Are you now ready to get the measures of position in a grouped data? To help you understand the next topic, notes with illustrative examples are provided. The Quartile for Grouped Data Recall that quartiles divide the distribution into four equal parts. The steps in computing the median are similar to that of Q1 and Q3. In finding the median, we first need to determine the median class. In the same manner, the Q1 and the Q3 class must be determined first before computing for the value of Q1 and Q3. The Q1 class is the class interval where the       th 4 N score is contained, while the class interval that contains the       3 th 4 N score is the Q3 class. In computing the quartiles of grouped data, the following formula is used: 4 b k Qk kN cf Q LB i f             where: LB = lower boundary of the Qk class N = total frequency = cumulative frequency of the class before the Qk class = frequency of the Qk class i = size of class interval k = nth quartile, where n = 1, 2, and 3 b cf All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
36. 36. D EPED C O PY 386 Example 1. Calculate the Q1, Q2, and Q3 of the Mathematics test scores of 50 students. Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6 Solution: Class Interval Scores Frequency (f) Lower Boundaries (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 N = 50 i Therefore, 25% of the students have a score less than or equal to 28.21. (28th-38th score) Q3 class (19th-27th score) Q2 class (7th-18th score) Q1 class 1 1 4 b Q N cf Q LB i f             1 12.5 6 25.5 5 12 Q         1 28.21Q   5  25.5LB  50N  6b cf  2 12Q f   50 4 4 12.5 N This means we need to find the class interval where the 12.5th score is contained. Note that the 7th-18th scores belong to the class interval: 26-30. So, the 12.5th score is also within the class interval. The Q1 class is class interval 26-30. Q1 class: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
37. 37. D EPED C O PY 387 Therefore, 50% of the students have a score less than or equal to 34.39 Therefore, 75% of the students have a score less than or equal to 40.27. The third quartile 40.27 falls within the class boundaries of 36-40 which is (35.5-40.5) Q2 class:  2 4 N  2 50 4 This means we need to find the class interval where the 25th score is contained. Note that the 19th-27th scores belong to the class interval: 31-35. So, the 25th score is also within the class interval. The Q2 class is the class interval 31-35. 3 3 3 4 b Q N cf Q LB i f             3 37.5 27 35.5 5 11 Q         3 40.27Q  Q3 class: 3 4 N This means we need to find the class interval where the 37.5th score is contained. Note that the 28th-38th scores belong to the class interval: 36-40. So, the 37.5th score is also within the class interval. The Q3 class is class interval 36-40. 2 2 2 4 b Q N cf Q LB i f             2 25 18 30.5 5 9 Q         2 34.39Q   30.5LB  50N  18b cf  2 9Q f  5i  35.5LB  50N  27b cf  2 11Q f  5i   100 4 25     3 50 4 150 4 37.5 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
38. 38. D EPED C O PY 388 The Deciles for Grouped Data Deciles are those values that divide the total frequency into 10 equal parts. The kth decile denoted by k D is computed as follows: 10 k b k D kN cf D LB i f              where: LB = lower boundary of the Dk class N = total frequency b cf = cumulative frequency before the Dk class kD f = frequency of the Dk class i = size of class interval k = nth decile where n = 1, 2, 3, 4, 5, 6, 7, 8, and 9 Example 2. Calculate the 7th decile of the Mathematics test scores of 50 students. Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
39. 39. D EPED C O PY 389 Solution: Class Interval Scores Frequency (f) Lower Boundaries (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 N = 50 Therefore, the 7th decile is equivalent to the 70th percentile. Therefore, 70% of the students got a score less than or equal to 39.14. (28th-38th score) D7 class 7 7 7 10 b D N cf D LB i f                      7 35 27 35.5 5 11 D 7 39.14D  D7 class: 7 10 N = = 350 10 = 35 This means we need to find the class interval where the 35th score is contained. Note that the 28th-38th scores belong to the class interval: 36-40. So, the 35th score is also within the class interval. The D7 class is the class interval 36-40. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
40. 40. D EPED C O PY 390 The Percentile for Grouped Data The percentile of grouped data is used to characterize values according to the percentage below them. Early on, you have already learned that kth quartile denoted by Qk and the kth deciles denoted by Dk are computed, respectively, as follows: 4 k b k Q kN cf Q LB i f             and 10 k b k D kN cf D LB i f              Finding percentiles of a grouped data is similar to that of finding quartiles and deciles of a grouped data. The kth percentile, denoted by Pk, is computed as follows: 100 k b k P kN cf P LB i f              where: LB = lower boundary of the kth percentile class N = total frequency b cf = cumulative frequency before the percentile class kP f = frequency of the percentile class i = size of class interval k = nth percentile where n = 1, 2, 3,…, 97, 98, and 99 Example 3. Calculate the 65th percentile and 32nd percentile of the Mathematics test scores of 50 students. Scores Frequency 46-50 4 41-45 8 36-40 11 31-35 9 26-30 12 21-25 6 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
41. 41. D EPED C O PY 391 (28th-38th score) P65 class (7th-18th score) Q1 class Solution: i = 5 Therefore, 65% of the students got a score less than or equal to 36-40. Class Interval Scores Frequency (f) Lower Boundaries (LB) Less than Cumulative Frequency (<cf) 46-50 4 45.5 50 41-45 8 40.5 46 36-40 11 35.5 38 31-35 9 30.5 27 26-30 12 25.5 18 21-25 6 20.5 6 65 65 65 100 b P N cf P LB i f              65 32.5 27 35.5 5 11 P         65 38P  P65 class : 65 100 N = = 3250 100 = 32.5 This means we need to find the class interval where the 32.5th score is contained. Note that the 28th-38th scores belong to the class interval: 36-40. So, the 32.5th score is also within the class interval. The P65 class is the class interval 36-40. 65 35.5 50 27 11 b P LB N Cf f     All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
42. 42. D EPED C O PY 392 Therefore, 32% of the students got a score less than or equal to 29-67. Percentile Rank Percentile ranks are particularly useful in relating individual scores to their positions in the entire group. A percentile rank is typically defined as the proportion of scores in a distribution that a specific score is greater than or equal to. For instance, if you received a score of 95 on a mathematics test and this score was greater than or equal to the scores of 88% of the students taking the test, then your percentile rank would be 88. An example is the National Career Assessment Examination (NCAE) given to Grade 9 students. The scores of students are represented by their percentile ranks.           100 P PR P P LB f P cf N i where: PR = percentile rank, the answer will be a percentage P cf = cumulative frequency of all the values below the critical value P = raw score or value for which one wants to find a percentile rank LB = lower boundary of the kth percentile class N = total frequency i = size of the class interval 32 32 32 100 b P N cf P LB i f              32 16 6 25.5 5 12 P         32 29.67P  P32 class: 32 100 N =  32 50 100 = 1600 100 = 16 This means we need to find the class interval where the 16th score is contained. Note that the 7th-18th scores belong to the class interval: 26-30. So, the 16th score is also within the class interval. The P32 class is class interval 26-30. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
43. 43. D EPED C O PY 393 Example 4. Find how many percent of the scores are greater than the cumulative frequency of 38 in the previous table. Solution: Scores Frequency cf 46-50 4 50 41-45 8 46 36-40 11 38 31-35 9 27 26-30 12 18 21-25 6 6           100 P PR P P LB f P cf N i Therefore, 65% of the scores are less than the cumulative frequency of 38, while 35% of the scores are greater than the cumulative frequency of 38. Example 5. Assume that a researcher wanted to know the percentage of consultants who made Php5,400 or more per day. Consultant Fees (in Php) Number of Consultants Cumulative Frequency 6400 – 7599 24 120 5200 – 6399 36 96 4000 – 5199 19 60 2800 – 3999 26 41 1600 – 2799 15 15 (28th – 38th score) 38 is within 36-40 LB = 35.5 P = 38 N = 50 P f = 11 P cf = 27 I = 5           38 35.5 27100 27 50 5PR P 65PR P  All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
44. 44. D EPED C O PY 394 Round off the resulting value to the nearest whole number. Therefore, 55% of consultants make Php 5,400.00 or less per day and 45% of consultants make Php 5,400.00 or more per day. Daily allowance of 60 students Class Interval f <cf 81-90 7 60 71-80 10 53 61-70 15 43 51-60 4 28 41-50 12 24 31-40 6 12 21-30 3 6 11-20 2 3 1-10 1 1 D6 P15 P35 D8 D4 P70 Q1 Q2 D8 Q3 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ Activity 3: Php 5,400 is within 5200-6399 LB = 5199.5 N = 120 P = 5,400.00 P cf = 60 P f = 36 i = 1200          5400 5199.5 36100 60 120 1200PR          100 P PR P P LB f P cf N i  55.01PR P All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
45. 45. D EPED C O PY 395 Given the frequency distribution, compute for each quantile and match it with the letter code of its corresponding value to complete the phrase in the preceding page: Q1 T. 43 Q2 I. 61.83 Q3 N. 72.5 P15 Y. 35.5 P35 L. 48 P70 A. 69.83 D6 M. 65.83 D4 C. 50/5 D8 O. 75.5 R. 34 The following is a distribution for the number of employees in 45 companies belonging to a certain industry. Calculate the third quartile, 85th percentile, and 4th decile of the number of employees given the number of companies. Number of Employees Number of Companies 41 – 45 11 36 – 40 6 31 – 35 9 26 – 30 7 21 – 25 8 16 – 20 4 Activity 4: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
46. 46. D EPED C O PY 396 Find the 1st quartile, 7th decile, 35th percentile, and percentile rank of 115 and 155 for the following distribution. Class Interval Frequency 151 – 160 8 141 – 150 12 131 – 140 6 121 – 130 10 111 – 120 7 101 – 110 11 91 – 100 13 81 – 90 9 71 – 80 4 After having several discussions, examples, and activities, you need to have a closer look once again if there are still aspects which you find confusing and hard. You are now ready to answer questions like: How can the position of data be described and used in solving real-life problems? Dennis and Christine scored 32 and 23, respectively, in the National Career Assessment Examination (NCAE). The determining factor for a college scholarship is that a student’s score should be in the top 10% of the scores of his/her graduating class. The students in the graduating class obtained the following scores in the NCAE. NCAE Scores f LB <cf 39 – 41 6 36 – 38 7 33 – 35 9 30 – 32 13 27 – 29 22 Activity 6: Activity 5: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.